| L(s) = 1 | + 7·3-s + 5·5-s + 35·7-s + 22·9-s − 11·11-s + 26·13-s + 35·15-s + 101·17-s − 127·19-s + 245·21-s + 58·23-s + 25·25-s − 35·27-s − 27·29-s + 177·31-s − 77·33-s + 175·35-s + 191·37-s + 182·39-s + 66·41-s − 444·43-s + 110·45-s − 2·47-s + 882·49-s + 707·51-s − 669·53-s − 55·55-s + ⋯ |
| L(s) = 1 | + 1.34·3-s + 0.447·5-s + 1.88·7-s + 0.814·9-s − 0.301·11-s + 0.554·13-s + 0.602·15-s + 1.44·17-s − 1.53·19-s + 2.54·21-s + 0.525·23-s + 1/5·25-s − 0.249·27-s − 0.172·29-s + 1.02·31-s − 0.406·33-s + 0.845·35-s + 0.848·37-s + 0.747·39-s + 0.251·41-s − 1.57·43-s + 0.364·45-s − 0.00620·47-s + 18/7·49-s + 1.94·51-s − 1.73·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.813480017\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.813480017\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 11 | \( 1 + p T \) |
| good | 3 | \( 1 - 7 T + p^{3} T^{2} \) |
| 7 | \( 1 - 5 p T + p^{3} T^{2} \) |
| 13 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 17 | \( 1 - 101 T + p^{3} T^{2} \) |
| 19 | \( 1 + 127 T + p^{3} T^{2} \) |
| 23 | \( 1 - 58 T + p^{3} T^{2} \) |
| 29 | \( 1 + 27 T + p^{3} T^{2} \) |
| 31 | \( 1 - 177 T + p^{3} T^{2} \) |
| 37 | \( 1 - 191 T + p^{3} T^{2} \) |
| 41 | \( 1 - 66 T + p^{3} T^{2} \) |
| 43 | \( 1 + 444 T + p^{3} T^{2} \) |
| 47 | \( 1 + 2 T + p^{3} T^{2} \) |
| 53 | \( 1 + 669 T + p^{3} T^{2} \) |
| 59 | \( 1 + 386 T + p^{3} T^{2} \) |
| 61 | \( 1 + 521 T + p^{3} T^{2} \) |
| 67 | \( 1 + 96 T + p^{3} T^{2} \) |
| 71 | \( 1 - 427 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1006 T + p^{3} T^{2} \) |
| 79 | \( 1 + 910 T + p^{3} T^{2} \) |
| 83 | \( 1 - 818 T + p^{3} T^{2} \) |
| 89 | \( 1 - 601 T + p^{3} T^{2} \) |
| 97 | \( 1 + 228 T + p^{3} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.594420748892667277919922177830, −8.690071383785601472897066802624, −8.095454676005905338365649227919, −7.70849359208453352277088172454, −6.31153232945241438440009905826, −5.18075322089660871503895958150, −4.33984730886222039599165240753, −3.16003240198432167781810149876, −2.11035386549301866449828241599, −1.30998572171616245883073442640,
1.30998572171616245883073442640, 2.11035386549301866449828241599, 3.16003240198432167781810149876, 4.33984730886222039599165240753, 5.18075322089660871503895958150, 6.31153232945241438440009905826, 7.70849359208453352277088172454, 8.095454676005905338365649227919, 8.690071383785601472897066802624, 9.594420748892667277919922177830
